# Eureka Math Algebra 2 Module 4 Lesson 28 Answer Key

## Engage NY Eureka Math Algebra 2 Module 4 Lesson 28 Answer Key

### Eureka Math Algebra 2 Module 4 Lesson 28 Exercise Answer Key

Exercise: Build the Helicopters

In preparation for your data collection, you will need to construct 20 paper helicopters following the blueprint given at the end of this lesson. For consistency, use the same type of paper for each helicopter. For greater stability, you may want to use a piece of tape to secure the two folded body panels to the body of the helicopter. By design, there will be some overlap from this folding in some helicopters. You will carry out an experiment to investigate the effect of wing length on flight time.

a. Construct 20 helicopters with wing length of 4 inches and body length of 3 inches. Label 10 each of these helicopters with the word long.

b. Take the other 10 helicopters, and cut 1 inch off each of the wings so that you have 10 helicopters with 3-inch wings. Label each of these helicopters with the word short.

c. How do you think wing length will affect flight time? Explain your answer.
Answers will vary. I think that a longer wing length will produce more resistance to the air, which will result in a longer flight time.

Exploratory Challenge 1: Data Collection

Once you have built the 20 helicopters, each of them will be flown by dropping the helicopter from a fixed distance above the ground (preferably 12 feet or higher – record this height for use when presenting your findings later). For consistency, drop all helicopters from the same height each time, and try to perform this exercise in a space where possible confounding factors such as wind gusts and drafts from heating and air conditioning are eliminated.

a. Place the 20 helicopters in a bag, shake the bag, and randomly pull out one helicopter. Drop the helicopter from the starting height and, using a stopwatch, record the amount of time it takes until the helicopter reaches the ground. Write down this flight time in the appropriate column in the table below. Repeat for the remaining 19 helicopters.

Some helicopters might fly more smoothly than others; you may want to record relevant comments in your report.

b. Why might it be important to randomize (impartially select) the order in which the helicopters were dropped?
(This is different from the randomization you perform later when you are allocating observations to groups to develop the randomization distribution.)
Randomizing the order in which the 20 helicopters are dropped will allow any peculiar, unforeseen, or unknown factors that may affect flight times to be distributed among the groups and not necessarily be localized to any one group.

Exploratory Challenge 2: Developing Claims and Using Technology

With the data in hand, you will now perform your analysis regarding the effect of wing length.

Experiment: Wing Length

In this experiment, you will examine whether wing length makes a difference in flight time. You will compare the helicopters with long wings (wing length of 4 inches, Group A) to the helicopters with short wings (wing length of 3 inches, Group B). Since you are dropping the helicopters from the same height in the same location, using the same type of paper, the only difference in the two groups will be the different wing lengths.

Questions: Does a 1-inch addition in wing length appear to result in a change in average flight time? If so, do helicopters with longer wing length or shorter wing length tend to have longer flight times on average?

Carry out a complete randomization test to answer these questions. Show all 5 steps, and use the Anova Shuffle applet described in the previous lessons to assist both in creating the distribution and with your computations. Be sure to write a final conclusion that clearly answers the questions in context.
Step 1 – Null hypothesis: A 1-inch increase in wing length does not change average flight time.
Alternative hypothesis: A 1-inch increase in wing length changes average flight time.

Step 2 – Students compute Diff from the experiment’s data. The value WILL most likely be statistically significant.

Step 3 – Randomization distribution of Diff developed by student using applet.

Step 4 – Compute the probability of obtaining a Duff more extreme than the value from the experiment. Since the original question is asking about a change in flight times (as opposed to a strict increase or decrease), the alternative hypothesis is of the form “different from,” and students should select the beyond choice from the applet under Count Samples.

Step 5 – While there are no specific criteria stated in the question for what is a small probability, students should consider probability values from previous work in determining “small” versus “not small.” Again, student values will vary. The important point is that students’ conclusions should be consistent with the probability value and their assessments of that value as follows:

→ If students deem the probability to be “small,” then they should state a conclusion based on a statistically significant result. More specifically, we support the claim that a 1-inch Increase in wing length changes average flight time. Given the sign of the observed difference and the method students have chosen for computing Diff (e.g., was it Group A’s mean minus Group B’s mean?), they should state as to whether the 1-inch increase in wing length appears to increase or decrease the average flight time.

→ If students deem the probability to be NOT “small,” then they should state a conclusion based on a result that is NOT statistically significant. More specifically, we DO NOT have evidence to support the claim that a 1 -inch increase in wing length changes average flight time.

The expectation is that the helicopters with longer wings should stay in flight longer on average than the helicopters with shorter wings.

### Eureka Math Algebra 2 Module 4 Lesson 28 Problem Set Answer Key

One other variable that can be adjusted in the paper helicopters is body width. See the blueprints for details.

Question 1.
Construct 10 helicopters using the blueprint from the lesson. Label each helicopter with the word narrow.

Question 2.
Develop a blueprint for a helicopter that is identical to the blueprint used in class except for the fact that the body width will now be 1.75 inches.

Question 3.
Use the blueprint to construct 10 of these new helicopters, and label each of these helicopters with the word wide.

Question 4.
Place the 20 helicopters in a bag, shake the bag, and randomly pull out one helicopter. Drop the helicopter from the starting height and, using a stopwatch, record the amount of time it takes until the helicopter reaches the ground. Write down this flight time in the appropriate column in the table below. Repeat for the remaining 19 helicopters.

Question 5.
Questions: Does a 0. 5-inch addition in body width appears to result in a change in average flight time? If so, do helicopters with wider body width (Group D) or narrower body width (Group C) tend to have longer flight times on average?
Carry out a complete randomization test to answer these questions. Show all 5 steps, and use the Anova Shuffle applet described in previous lessons to assist both in creating the distribution and with your computations. Be sure to write a final conclusion that clearly answers the questions in context.
Once the 20 helicopter flight time data are collected…
Step 1 – Null hypothesis: A 0.5-inch addition in body width does not change average flight time.
Alternative hypothesis: A 0.5-inch addition in body width changes average flight time.

Step 2 – Students compute Diff from the experiment’s data. The value WILL most likely be statistically significant.

Step 3 – Randomization distribution of Diff developed by student using applet.

Step 4 – Compute the probability of obtaining a Diff more extreme than the value from the experiment. Since the
original question is asking about a change in flight times (as opposed to a strict increase or decrease), the alternative hypothesis is of the form “different from,” and students should select the beyond choice from the applet under Count Samples.

Step 5 – While there are no specific criteria stated in the question for what is a small probability, students should consider probability values from previous work in determining “small” versus “not small.” Again, student values will vary. The important point is that students’ conclusions should be consistent with the probability value and their assessments of that value as follows:

→ If students deem the probability to be “small,” then they should state a conclusion based on a statistically significant result. More specifically, we support the claim that a 0.5-inch addition in body width changes average flight time.

Given the sign of the observed difference and the method students have chosen for computing Diff(e.g., was it Group D’s mean minus Group C’s mean?), they should state as to whether the 0.5-inch increase in body width appears to increase or decrease the average flight time.

→ If students deem the probability to be NOT “small,” then they should state a conclusion based on a result that is NOT statistically significant. More specifically, we DO NOT have evidence to support the claim that a 0. 5 -inch addition in body width changes average flight time.

The expectation is that the Group D helicopters (with wider body width) should have an average flight time that is significantly less than the average flight time of the control helicopters of Group C.

### Eureka Math Algebra 2 Module 4 Lesson 28 Exit Ticket Answer Key

Question 1.
Explain why you constructed a randomization distribution in order to decide if wing length has an effect on flight time.